Mathematical terminology

Invariant (mathematics)

In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects. (Wikipedia).

Invariant (mathematics)
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An introduction to Invariant Theory - Harm Derksen

Optimization, Complexity and Invariant Theory Topic: An introduction to Invariant Theory Speaker: Harm Derksen Affiliation: University of Michigan Date: June 4, 2018 For more videos, please visit http://video.ias.edu

From playlist Mathematics

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Commutative algebra 4 (Invariant theory)

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. This lecture is an informal historical summary of a few results of classical invariant theory, mainly to show just how complic

From playlist Commutative algebra

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What is a Symmetry?

Symmetries show up everywhere in physics. But what is a symmetry? While the symmetries of shapes can be interesting, a lot of times, we are more interested in symmetries of space or symmetries of spacetime. To describe these, we need to build "invariants" which give a mathematical represen

From playlist Relativity

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Introduction to geometric invariant theory 1: Noncommutative duality - Ankit Garg

Optimization, Complexity and Invariant Theory Topic: Introduction to geometric invariant theory 1: Noncommutative duality Speaker: Ankit Garg Affiliation: Microsoft Research New England Date: June 5. 2018 For more videos, please visit http://video.ias.edu

From playlist Mathematics

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Matrix invariants and algebraic complexity theory - Harm Derksen

Computer Science/Discrete Mathematics Seminar I Topic: Matrix invariants and algebraic complexity theory Speaker: Harm Derksen More videos on http://video.ias.edu

From playlist Mathematics

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Bertrand Eynard - An overview of the topological recursion

The "topological recursion" defines a double family of "invariants" $W_{g,n}$ associated to a "spectral curve" (which we shall define). The invariants $W_{g,n}$ are meromorphic $n$-forms defined by a universal recursion relation on $|\chi|=2g-2+n$, the initial terms $W_{0,1}$

From playlist ­­­­Physique mathématique des nombres de Hurwitz pour débutants

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Symmetry in Physics | Noether's theorem

▶ Topics ◀ Global / Local Symmetries, Continuous / Discrete Symmetries ▶ Social Media ◀ [Instagram] @prettymuchvideo ▶ Music ◀ TheFatRat - Fly Away feat. Anjulie https://open.spotify.com/track/1DfFHyrenAJbqsLcpRiOD9 If you want to help us get rid of ads on YouTube, you can support us on

From playlist Symmetry

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16. Invariants questions 27-29

This was an interesting experience. The first question I saw immediately how to do, as, at least for someone with the right background in elementary number theory, it was a genuinely straightforward question. The second looked pretty hard, but I happened to play around with it in a fruitfu

From playlist Thinking about maths problems in real time: mostly invariants problems

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12 Equivalence relations

Put all three properties of binary relations together and you have an equivalence relation.

From playlist Abstract algebra

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Inference: A Logical-Philosophical Perspective - Moderated Conversation w/ A.C. Paseau and Gila Sher

Inference:  A Logical-Philosophical Perspective. Moderated Conversation with Gila Sher, Department of Philosophy, University of California, San Diego on the talk by Alexander Paseau, Faculty of Philosophy, University of Oxford. The Franke Program in Science and the Humanities Understandi

From playlist Franke Program in Science and the Humanities

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Deep Maths - machine learning and mathematics

In December 2021 mathematicians at Oxford and Sydney universities together with their collaborators at Google DeepMind announced that they had successfully used tools from machine learning to discover new patterns in mathematics. But what exactly had they done and what are its implications

From playlist Oxford Mathematics Public Lectures

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Alex Davies, DeepMind | Machine learning with mathematicians

1/26/2022 New Technologies in Mathematics Seminar Speaker: Alex Davies, DeepMind Title: Machine learning with mathematicians Abstract: Can machine learning be a useful tool for research mathematicians? There are many examples of mathematicians pioneering new technologies to aid our u

From playlist Talks | AI for science

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Univalent foundations and the equivalence principle - Benedikt Ahrens

Short Talks by Postdoctoral Members Benedikt Ahrens - September 21, 2015 http://www.math.ias.edu/calendar/event/88134/1442858400/1442859300 More videos on http://video.ias.edu

From playlist Short Talks by Postdoctoral Members

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Sir Michael Atiyah - The Mysteries of Space [1991]

The 64th annual Gibbs Lecture was given by Sir Michael Atiyah, Fellow of the Royal Society, of Trinity College, Cambridge, England. At a conference in San Francisco, California in January 1991, he delivered "Physics and the mysteries of space", which was filmed and made available on videot

From playlist Mathematics

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Topological Strings and String Dualities (Lecture - 02) by Rajesh Gopakumar

J-Holomorphic Curves and Gromov-Witten Invariants DATE:25 December 2017 to 04 January 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex stru

From playlist J-Holomorphic Curves and Gromov-Witten Invariants

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A Lieb-Schultz-Mattis type theorem without continuous symmetry by Hal Tasaki

PROGRAM THERMALIZATION, MANY BODY LOCALIZATION AND HYDRODYNAMICS ORGANIZERS: Dmitry Abanin, Abhishek Dhar, François Huveneers, Takahiro Sagawa, Keiji Saito, Herbert Spohn and Hal Tasaki DATE : 11 November 2019 to 29 November 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore How do is

From playlist Thermalization, Many Body Localization And Hydrodynamics 2019

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Function Extraction: Mapping Programs into Mathematica Equations

Function Extraction™: Mapping Programs into Mathematica EquationsMuch of the trouble in software engineering stems from our inability to determine precisely the functionof a program; programmers routinely write programs whose precise function they do not know, and canonly speculate about;

From playlist Wolfram Technology Conference 2022

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Benedikt Ahrens - Univalent Foundations and the UniMath library - IPAM at UCLA

Recorded 13 February 2023. Benedikt Ahrens of Delft University of Technology presents "Univalent Foundations and the UniMath library" at IPAM's Machine Assisted Proofs Workshop. Abstract: Univalent Foundations (UF) were designed by Voevodsky as a foundation of mathematics that is "invarian

From playlist 2023 Machine Assisted Proofs Workshop

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11. Two more invariant problems

Part of a series of videos in which I (usually) solve problems in real time from a list of 60 problems that require the finding of invariants. Here I solve numbers 16 and 17. The first one I manage to do quite systematically. With the second, I see my way to the end fairly early on but it

From playlist Thinking about maths problems in real time: mostly invariants problems

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Group action | Lebesgue measure | Congruence (geometry) | Translational symmetry | Invariant (physics) | Topology | Angle | Invariant estimator | Conformal map | Complete set of invariants | Concurrent lines | Discrete mathematics | Total order | Invariant theory | Rotation | Multiplication | Element (mathematics) | Equivalence class | Change of basis | Square matrix | Isometry | Similarity (geometry) | Diameter | List of inequalities | Random variable | Subgroup | Complex number | Conical surface | Hoare logic | Differential geometry | Reflection (mathematics) | Plane (geometry) | Presentation of a group | Circle | Power set | Cross-ratio | Perimeter | Dynamical system | Homeomorphism | Fixed point (mathematics) | Symmetry | Trace (linear algebra) | Erlangen program | Orthogonal transformation | Predicate (mathematical logic) | Trigonometry | Screw axis | Knot invariant | Domain of a function | Mathematical constant | Algebra | Degree of a polynomial | Linear map | Scaling (geometry) | Number line | Addition | Correctness (computer science) | Equivalence relation | Geometry | Matrix (mathematics) | Symmetry in mathematics | Transformation (function) | Projective transformation | Inner automorphism | Mathematical object | Linear algebra | Distance | Invariant subspace | Polynomial | Gauss–Bonnet theorem | Congruence relation | Determinant | MU puzzle | Euler characteristic | Differentiable manifold | Variance | Invariants of tensors | Mathematics | Set (mathematics) | Spectrum of a matrix | Riemannian manifold | Algebraic geometry | Pi | Group theory | Normal subgroup | Loop invariant | Collinearity | Manifold | Differential invariant | Conic section | Invariant differential operator | Invariant measure | Topological space | Translation (geometry) | Absolute value | Operation (mathematics) | Finite set | Abstract interpretation | Group (mathematics) | Optimizing compiler | Circumference | Manifold decomposition | Formal methods | Counting | Cardinal number | Probability distribution | Subset | Ratio | Class (set theory) | Area | Triangle | List of mathematical identities | Rotation (mathematics)